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## G = C2×He3.4S3order 324 = 22·34

### Direct product of C2 and He3.4S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C2×He3.4S3
 Chief series C1 — C3 — C32 — C3×C9 — C9○He3 — He3.4S3 — C2×He3.4S3
 Lower central C3×C9 — C2×He3.4S3
 Upper central C1 — C2

Generators and relations for C2×He3.4S3
G = < a,b,c,d,e,f | a2=b3=c3=d3=f2=1, e3=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=bc-1, be=eb, fbf=b-1, cd=dc, ce=ec, fcf=c-1, de=ed, df=fd, fef=c-1e2 >

Subgroups: 457 in 93 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C9, C9, C9, C32, C32, D6, C2×C6, D9, C18, C18, C18, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C9, C3×C9, He3, 3- 1+2, 3- 1+2, D18, S3×C6, C2×C3⋊S3, C3×D9, C32⋊C6, C9⋊C6, C9⋊S3, C3×C18, C3×C18, C2×He3, C2×3- 1+2, C2×3- 1+2, C9○He3, C6×D9, C2×C32⋊C6, C2×C9⋊C6, C2×C9⋊S3, He3.4S3, C2×C9○He3, C2×He3.4S3
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, C3⋊S3, S3×C6, C2×C3⋊S3, C3×C3⋊S3, C6×C3⋊S3, He3.4S3, C2×He3.4S3

Smallest permutation representation of C2×He3.4S3
On 54 points
Generators in S54
(1 52)(2 53)(3 54)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 35)(11 36)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 41)(20 42)(21 43)(22 44)(23 45)(24 37)(25 38)(26 39)(27 40)
(1 42 33)(2 43 34)(3 44 35)(4 45 36)(5 37 28)(6 38 29)(7 39 30)(8 40 31)(9 41 32)(10 54 22)(11 46 23)(12 47 24)(13 48 25)(14 49 26)(15 50 27)(16 51 19)(17 52 20)(18 53 21)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)
(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)(28 34 31)(29 35 32)(30 36 33)(37 40 43)(38 41 44)(39 42 45)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27)(28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)
(1 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 54)(9 53)(10 40)(11 39)(12 38)(13 37)(14 45)(15 44)(16 43)(17 42)(18 41)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 36)(27 35)

G:=sub<Sym(54)| (1,52)(2,53)(3,54)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,35)(11,36)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,41)(20,42)(21,43)(22,44)(23,45)(24,37)(25,38)(26,39)(27,40), (1,42,33)(2,43,34)(3,44,35)(4,45,36)(5,37,28)(6,38,29)(7,39,30)(8,40,31)(9,41,32)(10,54,22)(11,46,23)(12,47,24)(13,48,25)(14,49,26)(15,50,27)(16,51,19)(17,52,20)(18,53,21), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,54)(9,53)(10,40)(11,39)(12,38)(13,37)(14,45)(15,44)(16,43)(17,42)(18,41)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,36)(27,35)>;

G:=Group( (1,52)(2,53)(3,54)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,35)(11,36)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,41)(20,42)(21,43)(22,44)(23,45)(24,37)(25,38)(26,39)(27,40), (1,42,33)(2,43,34)(3,44,35)(4,45,36)(5,37,28)(6,38,29)(7,39,30)(8,40,31)(9,41,32)(10,54,22)(11,46,23)(12,47,24)(13,48,25)(14,49,26)(15,50,27)(16,51,19)(17,52,20)(18,53,21), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,54)(9,53)(10,40)(11,39)(12,38)(13,37)(14,45)(15,44)(16,43)(17,42)(18,41)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,36)(27,35) );

G=PermutationGroup([[(1,52),(2,53),(3,54),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,35),(11,36),(12,28),(13,29),(14,30),(15,31),(16,32),(17,33),(18,34),(19,41),(20,42),(21,43),(22,44),(23,45),(24,37),(25,38),(26,39),(27,40)], [(1,42,33),(2,43,34),(3,44,35),(4,45,36),(5,37,28),(6,38,29),(7,39,30),(8,40,31),(9,41,32),(10,54,22),(11,46,23),(12,47,24),(13,48,25),(14,49,26),(15,50,27),(16,51,19),(17,52,20),(18,53,21)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54)], [(10,16,13),(11,17,14),(12,18,15),(19,22,25),(20,23,26),(21,24,27),(28,34,31),(29,35,32),(30,36,33),(37,40,43),(38,41,44),(39,42,45)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27),(28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54)], [(1,52),(2,51),(3,50),(4,49),(5,48),(6,47),(7,46),(8,54),(9,53),(10,40),(11,39),(12,38),(13,37),(14,45),(15,44),(16,43),(17,42),(18,41),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29),(25,28),(26,36),(27,35)]])

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 9A 9B 9C 9D ··· 9K 18A 18B 18C 18D ··· 18K order 1 2 2 2 3 3 3 3 3 3 6 6 6 6 6 6 6 6 6 6 9 9 9 9 ··· 9 18 18 18 18 ··· 18 size 1 1 27 27 2 3 3 6 6 6 2 3 3 6 6 6 27 27 27 27 2 2 2 6 ··· 6 2 2 2 6 ··· 6

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 6 6 type + + + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 S3 S3 D6 D6 D6 C3×S3 C3×S3 S3×C6 S3×C6 He3.4S3 C2×He3.4S3 kernel C2×He3.4S3 He3.4S3 C2×C9○He3 C2×C9⋊S3 C9⋊S3 C3×C18 C3×C18 C2×He3 C2×3- 1+2 C3×C9 He3 3- 1+2 C18 C3×C6 C9 C32 C2 C1 # reps 1 2 1 2 4 2 1 1 2 1 1 2 6 2 6 2 3 3

Matrix representation of C2×He3.4S3 in GL6(𝔽19)

 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18
,
 0 0 18 1 0 0 1 1 17 18 0 0 0 0 18 0 1 0 0 0 18 0 0 1 0 0 18 0 0 0 1 0 18 0 0 0
,
 0 18 0 0 0 0 1 18 0 0 0 0 1 0 18 18 0 0 0 18 1 0 0 0 1 0 0 0 18 18 0 18 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 18 18 0 0 1 1 0 0 18 18 0 0 0 0 1 0
,
 14 17 0 0 0 0 2 12 0 0 0 0 2 0 12 17 0 0 0 17 2 14 0 0 2 0 0 0 12 17 0 17 0 0 2 14
,
 0 18 0 0 0 0 18 0 0 0 0 0 0 0 0 0 18 0 18 18 0 0 1 1 0 0 18 0 0 0 18 18 1 1 0 0

G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[0,1,0,0,0,1,0,1,0,0,0,0,18,17,18,18,18,18,1,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,1,1,0,1,0,18,18,0,18,0,18,0,0,18,1,0,0,0,0,18,0,0,0,0,0,0,0,18,1,0,0,0,0,18,0],[1,0,0,1,1,0,0,1,0,1,1,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,18,1,0,0,0,0,18,0],[14,2,2,0,2,0,17,12,0,17,0,17,0,0,12,2,0,0,0,0,17,14,0,0,0,0,0,0,12,2,0,0,0,0,17,14],[0,18,0,18,0,18,18,0,0,18,0,18,0,0,0,0,18,1,0,0,0,0,0,1,0,0,18,1,0,0,0,0,0,1,0,0] >;

C2×He3.4S3 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3._4S_3
% in TeX

G:=Group("C2xHe3.4S3");
// GroupNames label

G:=SmallGroup(324,147);
// by ID

G=gap.SmallGroup(324,147);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,3171,453,2164,1096,7781]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=f^2=1,e^3=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,d*b*d^-1=b*c^-1,b*e=e*b,f*b*f=b^-1,c*d=d*c,c*e=e*c,f*c*f=c^-1,d*e=e*d,d*f=f*d,f*e*f=c^-1*e^2>;
// generators/relations

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